Graph and inverse function

Being the inverse of the exponential, the graph of the logarithm is obtained by reflecting that of the exponential, and it “inherits” all its features in mirror image.

Property — Features of the logarithm

For every $a > 0,\ a\ne 1$:

• Domain: $(0;+\infty)$;
• Range: $\mathbb{R}$;
• Monotonicity: increasing if $a>1$, decreasing if $0<a<1$;
• Zero: $\log_a 1 = 0$ (the curve passes through $(1;0)$);
• Asymptotes: the $y$-axis ($x=0$) is a vertical asymptote.

The graph of the logarithm is the reflection, with respect to the line $y=x$, of the graph of the corresponding exponential.

Mirror symmetry: the graph of $y=\log_2 x$ is the reflection of $y=2^x$ with respect to the line $y=x$. The point $(0;1)$ of the exponential corresponds to the point $(1;0)$ of the logarithm; the $x$-axis (horizontal asymptote of the exponential) becomes the $y$-axis (vertical asymptote of the logarithm).

In the following simulation you can change the base $a$ and observe how the logarithm and the exponential always remain mirror images with respect to the line $y=x$.

Drag the slider for the base $a$: the logarithm (green) and the exponential (dashed red) are mirror images with respect to the line $y=x$.

Links

Topics: Logarithmic function
Concepts: Inverse function · Graph of a function · Logarithm · Monotonicity
Functions: Exponential function · Logarithmic function
Skills: Interpreting a graph · Drawing a graph